Laplace Transform Calculator
Find Laplace transforms and inverse transforms with the table, the shifting theorem and partial fractions, and solve initial value problems, exactly and step by step.
About the Laplace Transform Calculator
The Laplace transform L{f(t)} = ∫₀^∞ e^(−st) f(t) dt turns derivatives into algebra: L{y′} = sY − y(0) and L{y″} = s²Y − s y(0) − y′(0). So a linear differential equation with its initial conditions becomes an equation for Y(s), which is solved and turned back into y(t) with the inverse transform.
Transforms are worked term by term from the table (L{tⁿ} = n!/sⁿ⁺¹, L{sin bt} = b/(s² + b²), …) with the first shifting theorem (multiplying by e^(at) replaces s by s − a) and multiplication by t (L{t f} = −F′(s)). The second shifting theorem handles unit steps: L{g(t)u(t − a)} = e^(−as)L{g(t + a)}, and L{δ(t − a)} = e^(−as). Inverses are found by partial fractions, or by the convolution theorem.
Every answer is checked: the algebra of each step, and the transform of the answer measured by numerical integration and compared with F(s). For an initial value problem the solution is also substituted back into the equation.
The formula
L{f(t)} = ∫₀^∞ e^(−st) f(t) dt; L{e^(at) f(t)} = F(s − a); L{t f(t)} = −F′(s)
- L{y′}
- sY(s) − y(0)
- L{y″}
- s²Y(s) − s y(0) − y′(0)
Worked example
y″ + 3y′ + 2y = 0, y(0) = 1, y′(0) = 0
- (s² + 3s + 2)Y = s + 3
- Y = (s + 3)/((s + 1)(s + 2)) = 2/(s + 1) − 1/(s + 2)
- y = 2e^(−t) − e^(−2t)
Questions people ask
Which functions are supported?
Sums of products of tⁿ, e^(at), sin(bt), cos(bt), sinh(bt) and cosh(bt) with fraction constants; terms switched on by a unit step u(t − a) or a Dirac delta δ(t − a); and inverses with a delay factor e^(−as).
How does the second shifting theorem work?
L{g(t)u(t − a)} = e^(−as) L{g(t + a)}: shift the function back by a, transform it, and multiply by e^(−as). Going back, L⁻¹{e^(−as)G(s)} = g(t − a)u(t − a). For the delta, L{δ(t − a)} = e^(−as).
What is the convolution theorem?
L⁻¹{F(s)G(s)} = ∫₀ᵗ f(τ)g(t − τ) dτ, the convolution (f * g)(t). The page finds f and g, writes the integral, gives its value (found by partial fractions of F·G) and checks it by evaluating the integral numerically.
Why must F(s) be a proper fraction?
A transform of an ordinary function goes to 0 as s grows. A polynomial part would need delta functions.